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M1L5w.txt
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#
# File: content-mit-8422-1x-captions/M1L5w.txt
#
# Captions for 8.422x module
#
# This file has 188 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
We have now three modes of light.
We would say, hey, a qubit.
Each qubit is two.
What is it?
Is it one and a half qubit?
Yes, it is, but we extend it to two qubits in a moment.
So what we simply have right now is
we have the mode c which controls the non-linear medium.
And we have what we have already discussed,
our interferometer with a phase shifter,
which is now the Kerr medium.
So I think even without doing any math,
we know that, by construction, when we have no photons in c,
that the modes a prime, b prime transform, become a and b,
because we have a balanced interferometer, which
is the identity transformation, unless we have a phase shift.
But when we have one photon in state c, then we actually swap.
So that's pretty cool.
A single photon in mode c will redirect the photon from a, b
to b, a.
So that's how one photon in state c, in mode c,
controls what happens to the photon in a, b.
OK.
So if we want a general description of that,
we obtain the output state from the input state
by using the matrix B. This was our interferometer based
on two beam splitters.
It involved mode a and b.
But now we throw in the phase shifter, which
is our Kerr operator, our non-linear phase shifter,
with b and c.
And this new addition to the interferometer
is described by e to the i non-linear susceptibility
L b dagger b c dagger c.
So let me give you one intermediate result.
We have now beam splitters on either side,
and that leads us to an expression.
Well, the beam splitters do nothing to the mode c,
but the beam splitters transform the modes
in linear combinations.
And if you do some tedious rearrangement of terms--
more details are given in the wiki--
you find that the essential result is now
the operator which we get is a dagger b minus b dagger
a over 2.
I'm not giving you the nonessential terms,
which are just phase shifts, global phase shifts.
And so this is just the result of the operator manipulation.
So if I would cover that, what is that?
Do you remember a dagger b minus b dagger a in the exponent?
Beam splitter.
It's a beam splitter.
And the beam splitter-- the beam splitter matrix
was sine theta cosine theta.
So we had a theta here.
So now you realize that this non-linear Mach-Zehnder
interferometer, the three optical elements
can be replaced by simply a single beam
splitter for mode a and b.
But the angle theta of the beam splitter
is now controlled by the photon field in state c.
So in other words, we have now a beam splitter.
I told you that a beam splitter is
nothing else than the rotation around the y-axis.
So now this non-linear Mach-Zehnder interferometer
is simply a beam splitter with the rotation
angle given by this term.
OK.
So this is the situation we just had.
And we can now get to two qubits by simply adding one more rail.
Remember, two rails with exactly one photon in it is a qubit.
So now we have a qubit here, and we have a qubit there.
And you see immediately one way how the upper qubit
acts on the lower qubit.
If the lower qubit has the photon in state c,
it flips the qubit in the other--
it flips the other qubit.
But if the upper qubit has the photon in d,
it doesn't do anything to the lower qubit.
So now we have a two-qubit operation.
So this is shown here.
But what I want to discuss now is
the situation when we throw in one more beam splitter.
And this beam splitter acts on the upper qubit.
Well, isn't it great how quickly we go from simple elements
to something which looks quite sophisticated?
So this device now is a simple optical device
which can entangle qubits, which leads to entanglement.
And actually, entanglement is our next big topic.
We want to talk about entanglement between particles,
entanglement between photons.
And by input using the Mach-Zehnder interferometer,
I was actually building up the situation,
which is extremely simple elements which
lead to entanglement.
We have everything for the description.
Each of those elements is described by a matrix,
which we have developed.
So therefore, by just multiplying those matrices,
you find immediately how this whole device is now
manipulating two qubits.
So if you want, you can just take the crank,
use what we have already done, and crank out
the result for it.
OK.
What I want to discuss with you now is what happens when we
take this device and we use exactly the states 01,
01 at the input, as I indicated.
So in terms of qubit language, I say
that the first qubit is in spin up,
and the lower qubit is also in spin up.
Spin down would mean that the single photon
is in the other state.
Our two-level system, our dual-rail, single-qubit
two-level system, is the photon can be either
in one of those states.
This one we call spin up.
This one we call spin down.
So what I want to discuss with you now what happens when we
start with this symmetric input up, up, which means 01, 01.
And since I don't want to scroll up and down so often,
everything is trivial beam splitters,
except for the situation when we have one photon each here.
And then the Kerr medium gives us a minus sign.
So that's what I want to put in.
So I want to develop the physics in 1, 2, 3 temporal steps.
The first temporal step is simply two beam splitters,
and this is shown here.
We start out, and now we have beam splitter
for the upper qubit, beam splitter for the lower qubit.
And well, by just multiplying it out,
we are now obtaining this summation state
for the four different modes.
And I'm dropping here factors of square root 2.
You can collect them at the end if you want.
And I said the only interesting situation
is where we have two photons, one each
in the two middle modes, because now the Kerr medium kicks
in and gives us a minus sign.
And now you can show by inspection that if you now
apply the beam splitter to modes a and b,
so you take 10 into a linear combination of 1 and 1,
but the beam splitter's only acting on the last two numbers
here, then you obtain a state which looks very simple.
So this is the output state of the device.
Any question?
And this output state is, if I use the spin language,
is an entangled state.
So now you have one simple example
how just using linear optics, very simple elements, you
can start with one photon here, one photon
here, a simple product state, and what comes out
is an entangled state.
Questions?
OK.
Let me tell you what I'm not telling you,
namely the wiki developed by Professor Ike Chuang
has now a wonderful section on a famous quantum algorithm,
the Deutsch-Jozsa algorithm.
Pretty much using the elements we discuss,
you can now realize the Deutsch-Jozsa algorithm,
which is one of the famous algorithms where
quantum logic, quantum computation,
is faster and more efficient than classical computation.
I decided not to present it to you
because it just leads to an even more complicated diagram
and more formulae.
You should just sit back and slowly read by it yourself.
There is no new idea introduced.
It's just the concepts we have discussed
can lead to very powerful algorithms.
What I want to rather do is I want to continue our discussion
and now talk about the object we have encountered here,
namely entangled states.